We note that the asymptotic analysis of the posterior distribution of the number of clusters for Dirichlet process mixtures has recently attracted considerable theoretical interest [1]}, [2]}, [3]}, and has motivated various methodological developments [4]}, [5]}. [2]} showed that, if \(\alpha \)  is sent deterministically to 0 at appropriate rates as \(n\rightarrow \infty \) , the posterior distribution of the number of clusters concentrates on finite values when data are generated from a finite mixture, which is a necessary condition for consistency. Such results are similar in spirit to ours, although we consider the substantially different setting where \(\alpha \)  is learned through a prior, which is arguably more natural in a Bayesian framework.
Finally, our results also provide an answer, at least partially, to the question of [1]}: “there exists a natural way to correct the problem instead of truncating the number of clusters?”, by showing that placing a prior on \(\alpha \)  can be sufficient to recover consistency.
